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Thread: Field Theory - Basic Theory - D&F Section 13.1

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    Super Member Bernhard's Avatar
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    Field Theory - Basic Theory - D&F Section 13.1

    I am reading Dummit and Foote (D&F) Section 13.1 Basic Theory of Field Extensions.

    I have a question regarding the nature of extension fields.

    Theorem 4 (D&F Section 13.1, page 513) states the following (see attachment):

    ---------------------------------------------------------------------------------------------------------------

    Theorem 4. Let $\displaystyle p(x) \in F[x] $ be an irreducible polynomial of degree n over a field F and let K be the field $\displaystyle F[x]/(p(x)) $. Let $\displaystyle \theta = x \ mod \ (p(x)) \in K $. Then the elements

    $\displaystyle 1, \theta, {\theta}^2, ... ... , {\theta}^{n-1} $

    are a basis for K as a vector space over F, so the degree of the extension is n i.e.

    $\displaystyle [K \ : \ F] = n $. Hence

    $\displaystyle K = \{ a_0 + a_1 \theta + a_2 {\theta}^2 + ... ... + a_{n-1} {\theta}^{n-1} \ | \ a_0, a_1, ... ... , a_{n-1} \in F \} $

    consists of all polynomials of degree $\displaystyle \lt n $ in $\displaystyle \theta $

    ------------------------------------------------------------------------------------------------------------------------------

    However, when we come to Example 4 on page 515 of D&F we read the following: (see attachment)

    (4) Let $\displaystyle F = \mathbb{Q} $ and $\displaystyle p(x) = x^3 - 2 $ which is irreducible by Eisenstein.

    Denoting a root of p(x) by $\displaystyle \theta $ we obtain the field

    $\displaystyle \mathbb{Q}[x]/(x^3 - 2) \cong \{a + b \theta + c {\theta}^2 \ | \ a, b, c \in \mathbb{Q} $

    with $\displaystyle {\theta}^3 = 2 $ an extension of degree 3. ... ... etc

    ------------------------------------------------------------------------------------------------------------------------------

    Now my problem is that in Theorem 4 we read


    $\displaystyle K = \{ a_0 + a_1 \theta + a_2 {\theta}^2 + ... ... + a_{n-1} {\theta}^{n-1} \ | \ a_0, a_1, ... ... , a_{n-1} \in F \} $ which becomes

    $\displaystyle K = \{a + b \theta + c {\theta}^2 $ in the situation of Example 4

    But then in Example 4 we have

    $\displaystyle K = \mathbb{Q}[x]/(x^3 - 2) \cong \{a + b \theta + c {\theta}^2 \ | \ a, b, c \in \mathbb{Q} $

    ???

    It seems that in Theorem 4, we have $\displaystyle \theta = x \ mod \ (p(x)) $ but in Example (4) we have $\displaystyle \theta = \sqrt[3]{2} $ and we do not have equality but only an isomorphism, that is $\displaystyle \mathbb{Q}[x]/(x^3 - 2) \cong \mathbb{Q}(\sqrt[3]{2} $.

    In Field theory we seem to prove that an irreducible polynomial has a root in a field that is isomorphic to the actual field that contains the root.

    Does what I am saying make sense? Can someone clarify this issue for me?

    Peter
    Last edited by Bernhard; Oct 1st 2013 at 05:40 PM.
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